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Question #41094

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix where each entry $a_{ij}$ ( $1 \le i, j \le 3$ ) is an independent random variable uniformly distributed in the interval $[-1, 2]$ . If $D = \det(A)$ , then the variance of $D$ is:

Options

A.
$\frac{81}{16}$
B.
$\frac{27}{4}$
C.
6
D.
$\frac{9}{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Random Variable Mapping|Determinant Product Rule|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question tests the interaction between determinant properties and the linearity of expectation.

The primary trap is 'Row/Column Exchange Parity'.

Students often incorrectly assume that the variance of a determinant is simply the sum of the variances of its individual permutation products (yielding 6), ignoring the covariance cross-terms.

Even when cross-terms are considered, students frequently overlook the negative signs introduced by odd permutations (transpositions), effectively calculating the variance of the permanent instead (yielding 27/4).

Additionally, a common oversight is failing to recognize that $E[\det(A)] = 0$ despite the mean of the entries being non-zero, due to the exact cancellation of symmetric positive and negative permutation expectations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Random Variable Mapping
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free